For homogeneous saturation, as *S* varies while porosity remains fixed,
the ratio does not change significantly until .At that point, increases dramatically and therefore decreases dramatically. Similarly, as , the only
changes in over most of the dynamic range of *S* are
in , which increases linearly with *S*. Then, when is almost at its
maximum value, increases dramatically, causing the
ratio to decrease dramatically. Thus, does not change monotonically with *S*, but first
increases a little and then decreases a lot.
These two ratios may be conveniently compared by plotting data from
various rocks and man-made porous media examples in the
(, )-plane [see Figure 1(b) and Figure 2].
We see that,
when data are collected at approximately equal intervals in *S*,
the low saturation points will all cluster together with nearly
constant
and small increases in , but the
final steps as lead to major decreases in both ratios.
The resulting plots appear as nearly straight lines in this plane,
with drained samples plotting to the upper right and fully saturated
samples plotting to the lower left in each of the examples shown in Figure 2.
The remaining ratio has the simplest behavior, since
increases monotonically in *S*, and does not change.
So is a monotonically increasing function of *S*, and
therefore can be considered a useful proxy of the saturation variable *S*.
[Compare Figures 1(c) and 1(d), and see Figure 3.]

Figure 2(a) includes the same sandstone data from Figure 1,
along with other sandstone data.
Similar data for five limestone samples (Cadoret *et al.*, 1998)
are plotted in Figure 2(b).
The straight line correlation of the data in the sandstone display
is clearly reconfirmed by the limestone data. Numerous other examples
of the correlation have been observed. [Fully dry and fully saturated
examples are shown here for some of these examples in Figures 2(c) and
2(d), for which partial saturation data were unavailable.]
No examples of appropriate
data for partially saturated samples have exhibited major deviations
from this behavior, although an extensive survey
of available data sets has been performed for materials including
limestones (Cadoret *et al.*, 1998),
sandstones (Murphy, 1984; Knight and Nolen-Hoeksema, 1990),
granites (Nur and Simmons, 1969), unconsolidated sands, and some
artificial materials such as ceramics and glass beads
(Berge *et al.*, 1995).
This straight line correlation is a very robust
feature of partial saturation data. The mathematical trick
that brings about this behavior will now be explained.

Consider the behavior as increases for fixed *S*.
Two of the parameters ( and )decrease as increases, but at different
rates, while the third () can have arbitrary variation.
[Recall (Bourbié *et al.*, 1987) that
rigorous bounds on the parameters are:
, ,, and .]
To understand the behavior on these plots in Figure 1 as changes,
it will prove convenient to consider polar coordinates (*r*,),
defined by

= w^2,
where *w* is an arbitrary scale factor with dimensions of velocity
(chosen so that *r* is a dimensionless radial coordinate for plots like
those in Figure 1). Now, if in addition we choose *w* to be
sufficiently large
so that *v*_{s}/*w* << 1 for typical values of *v*_{s} in our data sets, then,
using standard perturbation expansions, we have

r = w^2 (1 + v_s^4w^4)^12 w^2(1 + v_s^42 w^4) and

= ^-1(v_s^2w^2) v_s^2w^2.
Thus, the angle is well approximated by the ratio in
(thetasimeq), which depends only on the shear velocity *v*_{s}.
We know the shear velocity is a rather weak function of saturation
[*e.g.*, Figure 1(a)], but a much stronger function of porosity
[see, for example, Berge *et al.* (1995)].
So we see that the angle in these
plots is most strongly correlated with changes in the porosity.
In contrast, the radial position *r* is principally dependent on
the ratio , which we have already shown to be a strong
function of the saturation *S*, especially in the region close
to full liquid saturation. This analysis shows why the plots
in Figures 1(b) and 2 look the way they do and also why we might
be inclined to call these quasi-orthogonal (polar) plots of
saturation and porosity. Because of the function these plots play in
our analysis, we will call them the ``data-sorting'' plots.

In contrast, the plots in Figure 3 contain information about fluid spatial
distribution, as will be discussed at greater length later in this paper.
The bulk modulus *K*_{f} contains the only *S* dependence
in (Gassmann). Thus, for porous materials satisfying
Gassmann's homogeneous fluid conditions and for low enough frequencies,
the theory predicts that, if we use velocity data
in a two-dimensional plot with one axis being the saturation *S*
and the other being the ratio ,then the results will lie along an essentially straight (horizontal)
line until the saturation reaches (around 95% or higher),
where the curve formed by the data will quickly rise to
the value determined by the velocities at full liquid saturation.
On such a plot, the drained data appear in the lower left while
the fully saturated data appear in the upper right.
This behavior is illustrated in Figure 3(a) for Espeil limestone.
The behavior of the other plots in Figure 3 will be described below.

Before leaving this discussion of homogeneous saturation, we should
note that there is one laboratory saturation technique for which it is
known -- from direct observations (Cadoret *et al.*, 1998) using x-ray imaging -- that very
homogeneous liquid-gas mixtures will generally be produced. This
method is called ``depressurization.'' When such data are available
(see Figure 3), we expect they will always behave according to the
Gassmann-Domenico predictions. In contrast, the more common approach
which produces drainage data is less predictable, since the manner and
rate of drainage depend strongly on details of particular samples --
especially on surface energies that control capillarity and
on permeability magnitude and distribution. Thus, the drainage
technique can produce homogeneous saturation, or patchy saturation,
or anything in between.

4/28/2000